tag:blogger.com,1999:blog-32302427835369880802024-03-19T00:30:38.120-07:00QUADDo MathsQUADD WORLDhttp://www.blogger.com/profile/13254447413212520879noreply@blogger.comBlogger70125tag:blogger.com,1999:blog-3230242783536988080.post-73302531142435167962014-01-12T02:56:00.003-08:002014-01-12T02:56:33.152-08:00surface area and volume (part-2)<div dir="ltr" style="text-align: left;" trbidi="on">
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Unknownnoreply@blogger.com0tag:blogger.com,1999:blog-3230242783536988080.post-30243884409464331902014-01-12T01:27:00.002-08:002014-01-12T01:27:21.288-08:00SURFACE AREAS AND VOLUMES..CH-13<div dir="ltr" style="text-align: left;" trbidi="on">
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Unknownnoreply@blogger.com0tag:blogger.com,1999:blog-3230242783536988080.post-50837741998775396142013-12-22T01:05:00.000-08:002013-12-22T01:05:13.275-08:00circles....introduction<div dir="ltr" style="text-align: left;" trbidi="on">
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Unknownnoreply@blogger.com0tag:blogger.com,1999:blog-3230242783536988080.post-89697293307682878482013-12-11T05:30:00.000-08:002013-12-11T05:32:25.550-08:00MAJOR SEGMENT AND MINOR SEGMENT<div dir="ltr" style="text-align: left;" trbidi="on">
<br /></div>
<iframe width="560" height="315" src="//www.youtube.com/embed/w4OBwKd_cX0" frameborder="0" allowfullscreen></iframe>Unknownnoreply@blogger.com0tag:blogger.com,1999:blog-3230242783536988080.post-50386487554325574332013-12-01T02:19:00.000-08:002013-12-01T02:19:16.614-08:00probability(part 3 .....concept)<div dir="ltr" style="text-align: left;" trbidi="on">
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Unknownnoreply@blogger.com0tag:blogger.com,1999:blog-3230242783536988080.post-38376008382367745892013-12-01T02:15:00.002-08:002013-12-01T02:15:34.386-08:00probability(introduction)......................<div dir="ltr" style="text-align: left;" trbidi="on">
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Unknownnoreply@blogger.com0tag:blogger.com,1999:blog-3230242783536988080.post-85925421191238723822013-11-12T21:24:00.001-08:002013-11-12T21:24:10.444-08:00formulas used in coordinate geometry<div dir="ltr" style="text-align: left;" trbidi="on">
<div class="bodytext" style="font-family: Georgia, 'Times New Roman', Times, serif; font-size: 12px; line-height: 18px; text-align: justify;">
<strong>Distance between two points</strong></div>
<div class="bodytext" style="font-family: Georgia, 'Times New Roman', Times, serif; font-size: 12px; line-height: 18px; text-align: justify;">
Let P (x<sub>1</sub>, y<sub>1</sub>) and Q (x<sub>2</sub>, y<sub>2</sub>) be the two points. We have to find PQ.</div>
<div align="center" style="font-family: Georgia, 'Times New Roman', Times, serif;">
<img height="180" src="http://www.cbseguess.com/ebooks/x/maths/part2/images/image1209.gif" width="230" /></div>
<blockquote style="font-family: Georgia, 'Times New Roman', Times, serif;">
<img src="http://www.cbseguess.com/ebooks/x/maths/part2/images/image1211.gif" /><br />
<div class="bodytext" style="font-size: 12px; line-height: 18px; text-align: justify;">
<img height="14" src="http://www.cbseguess.com/ebooks/x/maths/part2/images/image1212.gif" width="15" />OM = x<sub>1</sub>, PM = y<sub>1</sub> = RN</div>
<div class="bodytext" style="font-size: 12px; line-height: 18px; text-align: justify;">
ON = x<sub>2</sub>, QN = y<sub>2</sub></div>
<div class="bodytext" style="font-size: 12px; line-height: 18px; text-align: justify;">
PR = MN = ON – OM</div>
<div class="bodytext" style="font-size: 12px; line-height: 18px; text-align: justify;">
= x<sub>2</sub> – x<sub>1</sub></div>
<div class="bodytext" style="font-size: 12px; line-height: 18px; text-align: justify;">
QR = QN – RN = y<sub>2</sub> – y<sub>1</sub></div>
<div class="bodytext" style="font-size: 12px; line-height: 18px; text-align: justify;">
By Pythagoras theorem</div>
<div class="bodytext" style="font-size: 12px; line-height: 18px; text-align: justify;">
PQ<sup>2</sup> = PR<sup>2</sup> + QR<sup>2</sup></div>
<div class="bodytext" style="font-size: 12px; line-height: 18px; text-align: justify;">
= (x<sub>2</sub> - x<sub>1</sub>)<sup>2</sup> + (y<sub>2</sub> – y<sub>1</sub>)<sup>2</sup></div>
<div class="bodytext" style="font-size: 12px; line-height: 18px; text-align: justify;">
<img height="30" src="http://www.cbseguess.com/ebooks/x/maths/part2/images/image1214.gif" width="207" /></div>
<div class="bodytext" style="font-size: 12px; line-height: 18px; text-align: justify;">
If x<sub>1</sub> = 0, y<sub>1</sub> = 0, x<sub>2</sub> = x and y<sub>2</sub> = y</div>
<div class="bodytext" style="font-size: 12px; line-height: 18px; text-align: justify;">
Then <img height="30" src="http://www.cbseguess.com/ebooks/x/maths/part2/images/image1216.gif" width="105" /></div>
</blockquote>
<div class="bodytext" style="font-family: Georgia, 'Times New Roman', Times, serif; font-size: 12px; line-height: 18px; text-align: justify;">
<strong>Section Formula</strong></div>
<div class="bodytext" style="font-family: Georgia, 'Times New Roman', Times, serif; font-size: 12px; line-height: 18px; text-align: justify;">
Let P (x, y) divided a line AB such that AP : PB = m<sub>1</sub> : m<sub>2</sub>.</div>
<div class="bodytext" style="font-family: Georgia, 'Times New Roman', Times, serif; font-size: 12px; line-height: 18px; text-align: justify;">
Let coordinates of A are (x<sub>1</sub>, y<sub>1</sub>) and B are (x<sub>2</sub>, y<sub>2</sub>).</div>
<blockquote style="font-family: Georgia, 'Times New Roman', Times, serif;">
<img height="233" src="http://www.cbseguess.com/ebooks/x/maths/part2/images/image1217.gif" width="245" /><br />
<img src="http://www.cbseguess.com/ebooks/x/maths/part2/images/image1219.gif" /><br />
<img src="http://www.cbseguess.com/ebooks/x/maths/part2/images/image1221.gif" /><br />
<div class="bodytext" style="font-size: 12px; line-height: 18px; text-align: justify;">
It is obvious that</div>
<img src="http://www.cbseguess.com/ebooks/x/maths/part2/images/image1223.gif" /><br />
<div class="bodytext" style="font-size: 12px; line-height: 18px; text-align: justify;">
Taking, <img src="http://www.cbseguess.com/ebooks/x/maths/part2/images/image1225.gif" /></div>
<img src="http://www.cbseguess.com/ebooks/x/maths/part2/images/image1227.gif" /><br />
<img src="http://www.cbseguess.com/ebooks/x/maths/part2/images/image1229.gif" /><br />
<div class="bodytext" style="font-size: 12px; line-height: 18px; text-align: justify;">
Similarity <img height="49" src="http://www.cbseguess.com/ebooks/x/maths/part2/images/image1231.gif" width="116" /></div>
<div class="bodytext" style="font-size: 12px; line-height: 18px; text-align: justify;">
Note</div>
<div class="bodytext" style="font-size: 12px; line-height: 18px; text-align: justify;">
(i) if P is mid point of AB, then AP : PB = 1 : 1</div>
<blockquote>
<img src="http://www.cbseguess.com/ebooks/x/maths/part2/images/image1233.gif" /></blockquote>
<div class="bodytext" style="font-size: 12px; line-height: 18px; text-align: justify;">
(ii) If m<sub>1</sub> : m<sub>2</sub> = k, then coordinates of P are <img src="http://www.cbseguess.com/ebooks/x/maths/part2/images/image1235.gif" /></div>
</blockquote>
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Unknownnoreply@blogger.com0tag:blogger.com,1999:blog-3230242783536988080.post-43669851362504600032013-11-09T05:25:00.000-08:002013-11-09T05:25:01.973-08:00coordinate geometry<div dir="ltr" style="text-align: left;" trbidi="on">
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Unknownnoreply@blogger.com0tag:blogger.com,1999:blog-3230242783536988080.post-74386817572591827912013-11-04T01:47:00.000-08:002013-11-04T01:47:00.328-08:00SAMPLE QUESTIONS -ARITHMETIC PROGRESSIONS<div dir="ltr" style="text-align: left;" trbidi="on">
1. An AP consists of 50 terms of which 3rd term is 12 and the last term is 106. Find the 29th<br />
term.<br />
Solution: 12 = a + 2d<br />
106 = a + 49d<br />
So, 106-12 = 47d<br />
Or, 94 = 47d<br />
Or, d = 2<br />
Hence, a = 8<br />
And, n29 = 8 + 28x2 = 64<br />
<div>
<br /></div>
<div>
<br /></div>
<div>
<div>
2. If the 3rd and the 9th terms of an AP are 4 and -8 respectively, which term of this AP is zero? </div>
<div>
Solution: -8 = a + 8d </div>
<div>
4 = a + 2d </div>
<div>
Or, -8 – 4 = 6d </div>
<div>
Or, -12 = 6d </div>
<div>
Or, d = -2 </div>
<div>
Hence, a = -8 + 16 = 8 </div>
<div>
0 = 8 + -2(n-1) </div>
<div>
Or, 8 = 2(n-1) </div>
<div>
Or, n-1 = 4 </div>
</div>
<div>
<br /></div>
<div>
<br /></div>
<div>
<div>
3. The 17th term of an AP exceeds its 10th term by 7. Find the common difference. </div>
<div>
Solution: n7 = a + 6d </div>
<div>
And, n10 = a + 9d </div>
<div>
Or, a + 9d – a – 6d = 7 </div>
<div>
Or, 3d = 7 </div>
<div>
Or, d = 7/3</div>
</div>
<div>
<br /></div>
<div>
<br /></div>
<div>
<br /></div>
<div>
<div>
4. Which term of the AP: 3. 15, 27, 39, … will be 132 more than its 54th term? </div>
<div>
Solution: d = 12, </div>
<div>
132/12 = 11 </div>
<div>
So, 54 + 11 = 65th term will be 132 more than the 54th term. </div>
</div>
<div>
<br /></div>
<div>
<br /></div>
<div>
<br /></div>
<div>
<div>
5. How many three digit numbers are divisible by 7? </div>
<div>
Solution: Smallest three digit number divisible by 7 is 105 </div>
<div>
Greatest three digit number divisible by 7 is 994 </div>
<div>
Number of terms </div>
<div>
= {(last term – first term )/common difference }+1 </div>
<div>
= {(994-105)/7}+1 </div>
<div>
= (889/7)+1=127+1=128 </div>
</div>
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<br /></div>
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<div>
WWW.WEEBLY.COM<SOURCE></div>
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Unknownnoreply@blogger.com0tag:blogger.com,1999:blog-3230242783536988080.post-83531970876135773132013-10-22T04:53:00.001-07:002013-10-22T04:53:14.678-07:00Arithematic progression<div dir="ltr" style="text-align: left;" trbidi="on">
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Unknownnoreply@blogger.com0tag:blogger.com,1999:blog-3230242783536988080.post-17562719700385459412013-10-19T08:27:00.002-07:002013-10-19T08:27:50.708-07:00METHODS TO SOLVE QUADRATIC EQUATIONS<div dir="ltr" style="text-align: left;" trbidi="on">
<div>
<div style="color: #cc0000;">
The solution of a quadratic equation is the value of x when you set the equation equal to zero <br />
i.e. When you solve the following general equation: ax² + bx + c=0
</div>
<blockquote style="color: #cc0000;">
Given a quadratic equation: ax ² + bx + c=0</blockquote>
<div style="color: #cc0000;">
There are three methods to solve a quadratic equation-</div>
<br />
<br />
First ------- Factorisation<br />
<blockquote>
<span style="background-color: #990000;"><span style="background-color: #990000;"></span><span style="background-color: #990000;"></span></span><span style="background-color: white;"></span>General Steps to solve by factoring
<ul>
<li> Create a factor chart for all factor pairs of c
<ul>
<li>A factor pair is just two numbers that multiply and give you 'c' </li>
</ul>
</li>
<li> Out of all of the factor pairs from step 1, look for the pair (if it exists) that add up to b
</li>
<li>Insert the pair you found in step 2 into two binomial.<a href="http://www.mathwarehouse.com/dictionary/B-words/binomial.html"> </a></li>
<li> Solve each binomial for zero to get the solutions of the quadratic equation.</li>
</ul>
</blockquote>
<div style="color: #cc0000;">
--Before learning next two steps we need to know that what is a discriminant??</div>
<div style="color: #cc0000;">
It is represented as D=b²-4ac</div>
<div style="color: #cc0000;">
It decides the nature of the roots of the quadratic equation.</div>
<div style="color: #cc0000;">
-if D is greater than 0 then roots are real</div>
<div style="color: #cc0000;">
-if D is o then roots are same</div>
<span style="color: #cc0000;"> -if D is less than 0 then roots are not real.</span><br />
<br />
Second----------Completing the squares<br />
General steps are-<br />
<ul style="text-align: left;">
<li> Divide the whole equation by the coefficient of x ² or 'a'</li>
</ul>
it becomes ax ²/a + bx/a + c/a=0<br />
or<br />
x ² + bx/a + c/a=0<br />
<ul style="text-align: left;">
<li>Now add and subtract (b/2a) ² to L.H.S.</li>
</ul>
Third----------Quadratic formula</div>
<div>
General steps</div>
<div>
<ul style="text-align: left;">
<li> Just apply the formula-</li>
</ul>
</div>
<div>
</div>
<div>
</div>
<div>
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<div>
<img alt="" src="data:image/png;base64,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" /></div>
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Unknownnoreply@blogger.com0tag:blogger.com,1999:blog-3230242783536988080.post-29956498243885524862013-10-06T01:20:00.001-07:002013-10-06T01:22:22.440-07:00quadratic equations (introduction)<div dir="ltr" style="text-align: left;" trbidi="on">
<iframe allowfullscreen="" frameborder="0" height="315" src="//www.youtube.com/embed/ViblRRhFamA" width="420"></iframe><br />
<div dir="ltr" style="text-align: left;" trbidi="on">
<br />
<br />
<span style="font-size: large;">In <a href="http://en.wikipedia.org/wiki/Elementary_algebra" title="Elementary algebra">elementary algebra</a>, a <b>quadratic equation</b> (from the <a href="http://en.wikipedia.org/wiki/Latin" title="Latin">Latin</a> <i>quadratus</i> for "<a href="http://en.wikipedia.org/wiki/Square_%28algebra%29" title="Square (algebra)">square</a>") is any equation having the form</span><br />
<span style="font-size: large;">
</span>
<br />
<dl><dd><span style="font-size: large;"><img alt="ax^2+bx+c=0" class="tex" src="http://upload.wikimedia.org/math/0/c/4/0c4913db725b72609d4825124dda12aa.png" /></span></dd></dl>
<span style="font-size: large;">
</span><span style="font-size: large;">where <i>x</i> represents an unknown, and <i>a</i>, <i>b</i>, and <i>c</i> are <a class="mw-redirect" href="http://en.wikipedia.org/wiki/Constant_term" title="Constant term">constants</a> with <i>a</i> not equal to 0. If <i>a</i> = 0, then the equation is <a href="http://en.wikipedia.org/wiki/Linear_equation" title="Linear equation">linear</a>, not quadratic. The constants <i>a</i>, <i>b</i>, and <i>c</i> are called, respectively, the quadratic <a href="http://en.wikipedia.org/wiki/Coefficient" title="Coefficient">coefficient</a>, the linear coefficient and the constant or free term.</span><br />
<br />
<span style="font-size: large;">source: youtube.com</span><br />
<br /></div>
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Unknownnoreply@blogger.com0tag:blogger.com,1999:blog-3230242783536988080.post-81990238401311246972013-09-09T17:33:00.001-07:002013-09-09T17:33:23.016-07:00THIS TIME GO FEARLESS TO CRACK YOUR MATHS EXAM!!<div dir="ltr" style="text-align: left;" trbidi="on">
<span style="font-family: Arial;"><span style="font-size: x-small;"><span style="color: maroon; font-size: x-large;"><strong>M</strong></span>any
students, even some really intelligent and talented ones, have a
strange enemy. They often find it difficult to finish the paper within
the allotted time. They are forced to leave a few questions just because
they run out of time and often it has been found that the questions
they leave are those which they otherwise could do very easily. It can
be very disappointing if you are forced to skip such easy questions. </span></span><br />
<span style="font-family: Arial;"><span style="font-size: x-small;">But how can you avoid a situation
like this? Many people suggest a single tablet for this "Time
management". But how to manage time and how to stop it from running out
is a difficult proposition, especially for an average 14-15-year-old
tenth grader. </span></span><br />
<span style="font-family: Arial; font-size: x-small;">Here are certain tips by experts from </span><span style="font-family: Arial;"><span style="font-size: x-small;">to help you out to finish your paper well before time.</span></span><br />
<br />
<span style="font-family: Arial;"><span style="font-size: x-small;">1. <strong>Understand your exam</strong></span></span><span style="font-family: Arial;"><span style="font-size: x-small;">The
most important thing is to understand the examination you are about to
take. In the class X mathematics paper, there are 30 questions in four
sections A, B, C and D and we have 180 minutes to answer these
questions. Here, a rough calculation is that we get about six minutes to
answer a question. <b>But that is not the fact.</b></span></span><br />
<span style="font-family: Arial;"><span style="font-size: x-small;">The question paper contains 'very
short answer' type, 'short answer' type and 'long answer' type questions
and the time requirement for each type is different. An ideal
allotment for the four sections is as shown below:</span></span><br />
<br />
<div class="MsoNormal" style="margin: 0in 0in 0pt;">
<br /></div>
<div class="MsoNormal" style="margin: 0in 0in 0pt;">
<span style="font-size: 11pt;"><span style="font-family: Arial;"><span style="font-size: x-small;">2. <b style="mso-bidi-font-weight: normal;">Use the first 15 minutes effectively</b></span></span></span></div>
<div class="MsoNormal" style="margin: 0in 0in 0pt;">
<span style="font-size: 11pt;"><span style="font-family: Arial;"><span style="font-size: x-small;">You
get a good 15 minutes in the beginning to read the question paper --
use this time to do just that, READ. Read all the 30 questions in 15
minutes. While reading, mark the questions into categories viz easy,
manageable and tough. This is done to have an overall idea about the
questions and make a rough plan.</span></span></span></div>
<div class="MsoNormal" style="margin: 0in 0in 0pt;">
<br /></div>
<div class="separator" style="clear: both; text-align: center;">
<a href="http://im.rediff.com/getahead/2008/feb/clx.gif" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img align="left" alt="" border="0" class="imgwidth" src="http://im.rediff.com/getahead/2008/feb/clx.gif" /></a></div>
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<div class="MsoNormal" style="margin: 0in 0in 0pt;">
<span style="font-size: 11pt;"><span style="font-family: Arial;"><span style="font-size: x-small;"> </span></span></span></div>
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<br /></div>
<div class="MsoNormal" style="margin: 0in 0in 0pt;">
<br /></div>
<div class="MsoNormal" style="margin: 0in 0in 0pt;">
<span style="font-size: 11pt;"><span style="font-family: Arial;"><span style="font-size: x-small;"> 3. <strong>Don't worry about the tough ones</strong></span></span></span></div>
<div class="MsoNormal" style="margin: 0in 0in 0pt;">
<span style="font-size: 11pt;"><span style="font-family: Arial;"><span style="font-size: x-small;">The
moment you find that there are a few tough questions; it is natural
that you start worrying about them. This is not required and will only
harm your performance.</span></span></span></div>
<div class="MsoNormal" style="margin: 0in 0in 0pt;">
<br /></div>
<div class="MsoNormal" style="margin: 0in 0in 0pt;">
<span style="font-size: 11pt;"><span style="font-family: Arial;"><span style="font-size: x-small;">The
fact is that they may look a bit tough on the surface, but when you
actually work on them you will find most of them to be much easier than
they seemed. So be happy about the easy ones and don't get unduly
worried about the tougher lot.</span></span></span></div>
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<br /></div>
<div class="MsoNormal" style="margin: 0in 0in 0pt;">
<span style="font-size: 11pt;"><span style="font-family: Arial;"><span style="font-size: x-small;">4. <strong>Prioritise your attempt</strong></span></span></span></div>
<div class="MsoNormal" style="margin: 0in 0in 0pt;">
<span style="font-size: 11pt;"><span style="font-family: Arial;"><span style="font-size: x-small;">Always
attempt the easy questions first and then move on to the manageable
ones and ensure that you complete them before taking on the difficult
ones. This will ensure that you are not leaving any question that you
know. </span></span></span></div>
<div class="MsoNormal" style="margin: 0in 0in 0pt;">
<br /></div>
<div class="MsoNormal" style="margin: 0in 0in 0pt;">
<span style="font-size: 11pt;"><span style="font-family: Arial;"><span style="font-size: x-small;">Once you successfully attempt all the easy and manageable questions, your confidence will grow and you will be</span></span></span></div>
<div class="MsoNormal" style="margin: 0in 0in 0pt;">
<span style="font-size: 11pt;"><span style="font-family: Arial;"><span style="font-size: x-small;">mentally ready to take on the more challenging questions.</span></span></span></div>
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<br /></div>
<div class="MsoNormal" style="margin: 0in 0in 0pt;">
<span style="font-size: 11pt;"><span style="font-family: Arial;"><span style="font-size: x-small;">5. <b style="mso-bidi-font-weight: normal;">Ensure speed and accuracy</b></span></span></span></div>
<div class="MsoNormal" style="margin: 0in 0in 0pt;">
<span style="font-size: 11pt;"><span style="font-family: Arial;"><span style="font-size: x-small;">Use
quicker methods in calculations to ensure that you are not wasting time
and your answers are correct. Mostly, we take a lot of time to solve a
problem if we happen to make some error in the process.</span></span></span></div>
<div class="MsoNormal" style="margin: 0in 0in 0pt;">
<br /></div>
<div class="MsoNormal" style="margin: 0in 0in 0pt;">
<span style="font-size: 11pt;"><span style="font-family: Arial; font-size: x-small;">For
example, if you make an error in the sign of a term (+/-), you may not
be able to solve questions involving quadratic equations or linear
equations. Therefore avoiding silly mistakes is very important to save time.</span><a class="" href="http://www.rediff.com/getahead/2008/jan/24board1.htm" target="new"><span style="font-family: Arial; font-size: x-small;"><br /></span></a></span></div>
<div class="MsoNormal" style="margin: 0in 0in 0pt;">
<br /></div>
<div class="MsoNormal" style="margin: 0in 0in 0pt;">
<span style="font-size: 11pt;"><span style="font-family: Arial;"><span style="font-size: x-small;">6.<strong> Keep an eye on your watch</strong></span></span></span></div>
<div class="MsoNormal" style="margin: 0in 0in 0pt;">
<span style="font-size: 11pt;"><span style="font-family: Arial;"><span style="font-size: x-small;">Keeping
an eye on your watch is of course not to increase your stress. This is
just to see that you are broadly adhering to the time allocation we
discussed in the beginning. A minor variation is not at all a reason to
worry.<span style="mso-spacerun: yes;"> </span></span></span></span></div>
<div class="MsoNormal" style="margin: 0in 0in 0pt;">
<b style="mso-bidi-font-weight: normal;"><span style="font-size: 11pt;"><span style="text-decoration: none;"></span></span></b></div>
<div class="MsoNormal" style="margin: 0in 0in 0pt;">
<br /></div>
<div class="MsoNormal" style="margin: 0in 0in 0pt;">
<span><span style="font-family: Arial; font-size: x-small;">7.<strong> Avoid thinking too much about a question</strong></span><span style="font-family: Arial; font-size: x-small;">Thinking
about the questions before you attempt them is essential; but not to
such an extent that you waste a lot of time on one question.</span><span style="mso-spacerun: yes;"><span style="font-family: Arial; font-size: x-small;"> </span></span></span></div>
<div class="MsoNormal" style="margin: 0in 0in 0pt;">
<br /></div>
<div class="MsoNormal" style="margin: 0in 0in 0pt;">
<span style="font-size: 11pt;"><span style="font-family: Arial;"><span style="font-size: x-small;">Also
you need not write a very lengthy answer to a question just because the
question is easy and you know it very well. Remember, you need to just
answer the question and nothing more. Any over-attempt will be a mere
waste of time.</span></span></span></div>
<div class="MsoNormal" style="margin: 0in 0in 0pt;">
<br /></div>
<span style="font-family: Arial; font-size: x-small;">Additionally,
you must practice the habit of finishing samples papers in 140-150
minutes. This will help you simulate and exercise examination pressures
better.</span> </div>
Unknownnoreply@blogger.com0tag:blogger.com,1999:blog-3230242783536988080.post-37685517624753078502013-09-08T02:24:00.003-07:002013-09-08T02:24:33.609-07:00TRIGNOMETRY-solve these.<div dir="ltr" style="text-align: left;" trbidi="on">
<h2 style="text-align: left;">
<span style="font-family: Arial; font-size: x-large;">QUESTIONS</span></h2>
<span style="font-family: Arial;"></span><br />
<li><span style="font-family: Arial;"> The area of a right triangle is 50. One of its angles is 45<sup>o</sup>. Find the lengths of the sides and hypotenuse of the triangle.
<hr />
</span></li>
<li><span style="font-family: Arial;">
In a right triangle ABC, tan(A) = 3/4. Find sin(A) and cos(A).
<hr />
</span></li>
<li><span style="font-family: Arial;">
In a right triangle ABC with angle A equal to 90<sup>o</sup>, find angle B and C so that sin(B) = cos(B).
<hr />
</span></li>
<li><span style="font-family: Arial;">
A rectangle has dimensions 10 cm by 5 cm. Determine the measures of the angles at the point where the diagonals intersect.
</span></li>
<span style="font-family: Arial;">
<hr />
<li>
The lengths of side AB and side BC of a scalene triangle ABC are 12 cm and 8 cm respectively. The size of angle C is 59<sup>o</sup>. Find the length of side AC.
</li>
<hr />
<li>
From the top of a 200 meters high building, the angle of depression to
the bottom of a second building is 20 degrees. From the same point, the
angle of elevation to the top of the second building is 10 degrees.
Calculate the height of the second building.
</li>
<hr />
<li>
Karla is riding vertically in a hot air balloon, directly over a point P
on the ground. Karla spots a parked car on the ground at an angle of
depression of 30<sup>o</sup>. The balloon rises 50 meters. Now the angle of depression to the car is 35 degrees. How far is the car from point P?
</li>
<hr />
</span><br />
<br /><span style="font-family: Arial;"></span>
<li style="text-align: left;"><span style="font-family: Arial;">
If the shadow of a building increases by 10 meters when the angle of elevation of the sun rays decreases from 70<sup>o</sup> to 60<sup>o</sup>, what is the height of the building? </span></li>
<br /><br />
<br />
<br />
<h2 style="text-align: left;">
<span style="font-size: x-large;"> ANSWERS</span></h2>
<br />
<br />
<br />
<li><span style="font-family: Arial;"><a href="http://www.blogger.com/blogger.g?blogID=3230242783536988080" name="solutions">
The triangle is right and the size one of its angles is 45<sup>o</sup>; the third angle has a size 45<sup>o</sup>
and therefore the triangle is right and isosceles. Let x be the length
of one of the sides and H be the length of the hypotenuse.
<br />
<br />
Area = (1/2)x<sup>2</sup> = 50 , solve for x: x = 10
<br />
<br />
We now use Pythagoras to find H: x<sup>2</sup> + x<sup>2</sup> = H<sup>2</sup>
<br />
<br />
Solve for H: H = 10 sqrt(2)
</a></span></li>
<span style="font-family: Arial;"><a href="http://www.blogger.com/blogger.g?blogID=3230242783536988080" name="solutions">
<hr />
<li>
Let a be the length of the side opposite angle A, b the length of the
side adjacent to angle A and h be the length of the hypotenuse.
<br />
<br />
tan(A) = opposite side / adjacent side = a/b = 3/4
<br />
<br />
We can say that: a = 3k and b = 4k , where k is a coefficient of proportionality. Let us find h.
<br />
<br />
Pythagoras theorem: h<sup>2</sup> = (3k)<sup>2</sup> + (5k)<sup>2</sup>
<br />
<br />
Solve for h: h = 5k
<br />
<br />
sin(A) = a / h = 3k / 5k = 3/5 and cos(A) = 4k / 5k = 4/5
</li>
<hr />
<li>
Let b be the length of the side opposite angle B and c the length of the
side opposite angle C and h the length of the hypotenuse.
<br />
<br />
sin(B) = b/h and cos(B) = c/h
<br />
<br />
sin(B) = cos(B) means b/h = c/h which gives c = b
<br />
<br />
The two sides are equal in length means that the triangle is isosceles and angles B and C are equal in size of 45<sup>o</sup>.
</li>
</a></span><br />
<br />
<br />
<br />
<li><span style="font-family: Arial;"><a href="http://www.blogger.com/blogger.g?blogID=3230242783536988080" name="solutions">
The diagram below shows the rectangle with the diagonals and half one of the angles with size x.
<br />
<br />
tan(x) = 5/2.5 = 2 , x = arctan(2)
<br />
<br />
larger angle made by diagonals 2x = 2 arctan(2) = 127<sup>o</sup> (3 significant digits)
<br />
<br />
Smaller angle made by diagonals 180 - 2x = 53<sup>o</sup>.
<div align="center">
<img alt="solution problem diagonals of rectangle" height="222" src="http://www.analyzemath.com/high_school_math/grade_10/graphs/trigonometry_g10_6.gif" width="313" />
</div>
</a></span></li>
<span style="font-family: Arial;"><a href="http://www.blogger.com/blogger.g?blogID=3230242783536988080" name="solutions">
<hr />
<li>
Let x be the length of side AC. Use the cosine law
<br />
<br />
12<sup>2</sup> = 8<sup>2</sup> + x<sup>2</sup> - 2*8*x*cos(59<sup>o</sup>)
<br />
<br />
Solve the quadratic equation for x: x = 14.0 and x = -5.7
<br />
<br />
x cannot be negative and therefore the solution is x = 14.0 (rounded to one decimal place).
</li>
</a></span><br />
<br />
<br />
<br />
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" /><br />
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Unknownnoreply@blogger.com0tag:blogger.com,1999:blog-3230242783536988080.post-65103042589745777182013-08-02T03:56:00.000-07:002013-08-02T03:56:03.712-07:00AREAS OF SIMILAR TRIANGLES(PROOF)<div dir="ltr" style="text-align: left;" trbidi="on">
<img src="http://www.meritnation.com/img/userimages/mn_images/image/proof%20of%20the%20theorem___1.png" /><br />
<img src="http://www.meritnation.com/img/userimages/mn_images/image/proof%20of%20the%20theorem%20___2.png" /><br />
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Unknownnoreply@blogger.com0tag:blogger.com,1999:blog-3230242783536988080.post-67219629868267062712013-07-23T09:05:00.002-07:002013-07-23T09:05:33.881-07:00BASIC PROPORTIONALITY THEOREM<div dir="ltr" style="text-align: left;" trbidi="on">
<iframe allowfullscreen="" frameborder="0" height="315" src="//www.youtube.com/embed/4EKk65oPv-g" width="420"></iframe><br />
<div dir="ltr" style="text-align: left;" trbidi="on">
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</div>
Unknownnoreply@blogger.com0tag:blogger.com,1999:blog-3230242783536988080.post-46792793623878642132013-07-22T05:48:00.002-07:002013-07-22T05:52:15.813-07:00PROVING CONVERSE OF BPT THEORM<div dir="ltr" style="text-align: left;" trbidi="on">
<u><b>Statement<br /> If a line divides any two sides of a triangle (Δ) in the same ratio, then the line must be parallel (||) to the third side. <br />Given<br /> In ΔABC, D and E are the two points of AB and AC respectively, <br /> such that, AD/DB = AE/EC. <br /><br />To Prove<br /> DE || BC <br /><br />Proof<br /> In ΔABC,<br /> given, AD/DB = AE/EC ----- (1)<br /><br /> Let us assume that in ΔABC, the point F is an intersect on the side AC.<br /> So we can apply the Thales Theorem, AD/DB = AF/FC ----- (2)<br /><br /> Simplify, in (1) and (2) ==> AE/EC = AF/FC <br /> Add 1 on both sides, ==> (AE/EC) + 1 = (AF/FC) + 1 <br /> ==> (AE+EC)/EC = (AF+FC)/FC <br /> ==> AC/EC = AC/FC <br /> ==> EC = FC <br /> From the above, we can say that the points E and F coincide on AC. <br /> i.e., DF coincides with DE. <br /><br /> Since DF is parallel to BC, DE is also parallel BC <br /><br /> Hence the Converse of Basic Proportionality theorem is proved.</b></u><br />
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<u><b>SOURCE- EASYCALCULATION.COM</b></u></div>
Unknownnoreply@blogger.com0tag:blogger.com,1999:blog-3230242783536988080.post-1616919870291027522013-07-21T04:49:00.002-07:002013-07-21T04:49:53.314-07:00similarity in triangles<div dir="ltr" style="text-align: left;" trbidi="on">
<span style="font-size: large;">Two geometrical objects are called <b>similar</b> if they both have the same shape, or one has the same shape as the mirror image of the other. More precisely, one can be obtained from the other by uniformly scaling (enlarging or shrinking), possibly with additional translation, rotation and reflection<a href="http://en.wikipedia.org/wiki/Reflection_%28mathematics%29" title="Reflection (mathematics)"></a>.
This means that either object can be rescaled, repositioned, and
reflected, so as to coincide precisely with the other object. If two
objects are similar, each is congruent to the result of a uniform scaling of the other.</span><br />
<span style="font-size: large;">
</span><span style="font-size: large;"><br /></span>
<span style="font-size: large;">
</span><span style="font-size: large;">If two angles of a triangle have measures equal to the measures of
two angles of another triangle, then the triangles are similar.
Corresponding sides of similar polygons are in proportion, and
corresponding angles of similar polygons have the same measure.</span><br />
<span style="font-size: large;">
</span><span style="font-size: large;">This article assumes that a scaling can have a scale factor of 1, so
that all congruent shapes are also similar, but some school text books
specifically exclude congruent triangles from their definition of
similar triangles by insisting that the sizes must be different to
qualify as similar.</span><br />
<span style="font-size: large;"> <table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody>
<tr><td style="text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiUtaVjDKNRJfAAo4tZSKEG9hW49657bu_qDauEDTWELioiZe6jJFs5PCc5ecp6RHwGp8RpUCtoGohUMGBn4q1SR2vfjHhrSLcuOTcgYAASfo1RRccFok1muilO-A-V9TnFgrqC8QCwZ-8/s1600/9.png" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img border="0" height="221" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiUtaVjDKNRJfAAo4tZSKEG9hW49657bu_qDauEDTWELioiZe6jJFs5PCc5ecp6RHwGp8RpUCtoGohUMGBn4q1SR2vfjHhrSLcuOTcgYAASfo1RRccFok1muilO-A-V9TnFgrqC8QCwZ-8/s320/9.png" width="320" /></a></td></tr>
<tr><td class="tr-caption" style="text-align: center;">figures with same colours are similar</td><td class="tr-caption" style="text-align: center;"><br /></td></tr>
</tbody></table>
</span><br />
<br />
<h2>
<span class="mw-headline" id="Similar_triangles">Similar triangles</span></h2>
<span style="font-size: large;">Two triangles <img alt="\triangle ABC" class="tex" src="http://upload.wikimedia.org/math/6/3/5/635759e23aaf6e02541e3b72d65268d0.png" /> and <img alt="\triangle DEF" class="tex" src="http://upload.wikimedia.org/math/3/e/3/3e39eb121ee9eb5a4106ef376cb1df52.png" /> are said to be similar if either of the following equivalent conditions holds:</span><br />
<span style="font-size: large;">
</span><span style="font-size: large;">1. They have two identical angles, which implies that their angles are all identical. For instance:</span><br />
<span style="font-size: large;">
</span>
<dl><dd><span style="font-size: large;"><img alt=" \angle BAC" class="tex" src="http://upload.wikimedia.org/math/b/8/d/b8d7a7847dcba2f1d40731cfeb648e9f.png" /> is equal in measure to <img alt="\angle EDF" class="tex" src="http://upload.wikimedia.org/math/f/0/8/f0849440484778156922ed9e3ee2d980.png" />, and <img alt="\angle ABC" class="tex" src="http://upload.wikimedia.org/math/7/0/c/70c612060bb4c336ea559881305cfcaf.png" /> is equal in measure to <img alt="\angle DEF" class="tex" src="http://upload.wikimedia.org/math/4/e/1/4e143e84a7a5356e818cdffa5b52652c.png" />. This also implies that <img alt="\angle ACB" class="tex" src="http://upload.wikimedia.org/math/f/3/8/f3867d8b3f8a94e03588997727f47c94.png" /> is equal in measure to <img alt="\angle DFE" class="tex" src="http://upload.wikimedia.org/math/4/1/9/419c7b7b7ebf0082443f378f80f8851a.png" />.</span></dd></dl>
<span style="font-size: large;">
</span><span style="font-size: large;">2. Corresponding sides have lengths in the same ratio:</span><br />
<span style="font-size: large;">
</span>
<dl><dd><span style="font-size: large;"><img alt=" {AB \over DE} = {BC \over EF} = {AC \over DF}" class="tex" src="http://upload.wikimedia.org/math/8/5/e/85ecce30d1e01aa37a010beebd7571f0.png" />. This is equivalent to saying that one triangle (or its mirror image) is an enlargement of the other.</span></dd></dl>
<span style="font-size: large;">
</span><span style="font-size: large;">3. Two sides have lengths in the same ratio, and the angles included between these sides have the same measure. For instance:</span><br />
<span style="font-size: large;">
</span>
<dl><dd><span style="font-size: large;"><img alt=" {AB \over DE} = {BC \over EF} " class="tex" src="http://upload.wikimedia.org/math/4/3/7/437f78cd74c20a811fb0e2f22671af0e.png" /> and <img alt="\angle ABC" class="tex" src="http://upload.wikimedia.org/math/7/0/c/70c612060bb4c336ea559881305cfcaf.png" /> is equal in measure to <img alt="\angle DEF" class="tex" src="http://upload.wikimedia.org/math/4/e/1/4e143e84a7a5356e818cdffa5b52652c.png" />.</span></dd></dl>
<span style="font-size: large;">
</span><span style="font-size: large;">When two triangles <img alt="\triangle ABC" class="tex" src="http://upload.wikimedia.org/math/6/3/5/635759e23aaf6e02541e3b72d65268d0.png" /> and <img alt="\triangle DEF" class="tex" src="http://upload.wikimedia.org/math/3/e/3/3e39eb121ee9eb5a4106ef376cb1df52.png" /> are similar, one writes</span><br />
<span style="font-size: large;">
</span>
<dl><dd><span style="font-size: large;"><img alt="\triangle ABC\sim\triangle DEF \, " class="tex" src="http://upload.wikimedia.org/math/5/a/2/5a22de4c59b832ed25006108b7d574db.png" /></span></dd></dl>
<span style="font-size: large;">
</span><span style="font-size: large;">or</span><br />
<span style="font-size: large;">
</span>
<dl><dd><span style="font-size: large;"></span></dd><dd><span style="font-size: large;"></span></dd><dd><span style="font-size: large;"></span></dd><dd><span style="font-size: large;"></span></dd><dd><span style="font-size: large;"><img alt="\triangle ABC \, ||| \,\triangle DEF \, " class="tex" src="http://upload.wikimedia.org/math/9/2/c/92c455e8c338e0fa9e35d4e01653e03a.png" /></span></dd></dl>
<span style="font-size: large;"> source: wikipedia.org </span></div>
Unknownnoreply@blogger.com0tag:blogger.com,1999:blog-3230242783536988080.post-16347334257046985302013-07-13T02:34:00.001-07:002013-07-13T02:34:58.824-07:00OGIVES<div dir="ltr" style="text-align: left;" trbidi="on">
<h1 class="documentFirstHeading" style="background-color: white; border-bottom-color: rgb(221, 221, 221); border-bottom-style: solid; border-bottom-width: 2px; clear: both; color: #111111; font-family: 'Helvetica Neue', Arial, Helvetica, sans-serif; font-size: 29px; line-height: 36px; margin: 4px 0px 0px; padding: 0px; text-shadow: rgb(221, 221, 221) 0px 2px 0px;">
<br /></h1>
<div class="documentDescription" style="background-color: white; color: #999999; font-family: 'Helvetica Neue', Arial, Helvetica, sans-serif; font-size: 1.25em; font-weight: bold; margin-bottom: 1em; padding: 0px;">
</div>
<div id="content-core" style="background-color: white; font-family: 'Helvetica Neue', Arial, Helvetica, sans-serif; font-size: 19px; margin: 0px; padding: 0px;">
<div class="xmlTextWidget xmltext-field" id="form-widgets-body" style="line-height: 1.55em; margin: 1.2em 0px; padding: 0px;">
<div style="line-height: 1.55em; margin: 1.2em 0px; padding: 0px;">
<div id="cnx_main" style="line-height: 1.55em; margin: 1.2em 0px; padding: 0px;" xmlns:bib="http://bibtexml.sf.net/" xmlns:cnx="http://cnx.rice.edu/cnxml" xmlns:md="http://cnx.rice.edu/mdml/0.4" xmlns:mod="http://cnx.rice.edu/#moduleIds" xmlns:qml="http://cnx.rice.edu/qml/1.0" xmlns="http://www.w3.org/1999/xhtml">
<div class="para" style="line-height: 1.55em; margin-bottom: 1.2em; margin-top: 1.2em; padding: 0px;">
Cumulative histograms, also known as ogives, are graphs that can be used to determine how many data values lie above or below a particular value in a data set. The cumulative frequency is calculated from a frequency table, by adding each frequency to the total of the frequencies of all data values before it in the data set. The last value for the cumulative frequency will always be equal to the total number of data values, since all frequencies will already have been added to the previous total.</div>
<div class="para" style="line-height: 1.55em; margin-bottom: 1.2em; margin-top: 1.2em; padding: 0px;">
An ogive is drawn by</div>
<div class="list" style="line-height: 1.55em; margin: 1.2em 0px; padding: 0px;">
<ul class="bullet" style="list-style-image: initial; list-style-position: initial; margin: 0px 0px 0px 1.5em; padding: 0px;">
<li class="item" style="line-height: 1.5em; margin: 0px 0px 0.25em; padding: 0px 0px 0.5em;">plotting the beginning of the first interval at a <img src="http://m.everythingmaths.co.za/@@mobile_image?key=85067ce783e2f89ced535d722b824af5.png" style="border: 0px; margin: 0px; padding: 0px; vertical-align: middle;" />-value of zero;</li>
<li class="item" style="line-height: 1.5em; margin: 0px 0px 0.25em; padding: 0px 0px 0.5em;">plotting the end of every interval at the <img src="http://m.everythingmaths.co.za/@@mobile_image?key=85067ce783e2f89ced535d722b824af5.png" style="border: 0px; margin: 0px; padding: 0px; vertical-align: middle;" />-value equal to the cumulative count for that interval; and</li>
<li class="item" style="line-height: 1.5em; margin: 0px 0px 0.25em; padding: 0px 0px 0.5em;">connecting the points on the plot with straight lines.</li>
</ul>
</div>
In this way, the end of the final interval will always be at the total number of data since we will have added up across all intervals.</div>
<div id="cnx_main" style="line-height: 1.55em; margin: 1.2em 0px; padding: 0px;" xmlns:bib="http://bibtexml.sf.net/" xmlns:cnx="http://cnx.rice.edu/cnxml" xmlns:md="http://cnx.rice.edu/mdml/0.4" xmlns:mod="http://cnx.rice.edu/#moduleIds" xmlns:qml="http://cnx.rice.edu/qml/1.0" xmlns="http://www.w3.org/1999/xhtml">
<h2 class="example-header" style="color: #111111; font-size: 1.1em; line-height: 29.75px; margin: 0px 0px 0.5em; padding: 0px;">
<span class="cnx_label" style="margin: 0px; padding: 0px;">Example 1: </span><strong class="title" style="margin: 0px; padding: 0px;">Cumulative frequencies and ogives</strong></h2>
<div class="problem" style="line-height: 29.75px; margin: 1.2em 0px; padding: 0px;">
<div class="para" style="line-height: 1.55em; margin-bottom: 1.2em; margin-top: 1.2em; padding: 0px;">
Determine the cumulative frequencies of the following grouped data and complete the table below. Use the table to draw an ogive of the data.</div>
<table align="center" cellpadding="0" cellspacing="0" style="border-collapse: collapse; border-spacing: 0px; border: 1px solid; color: black; font-size: 19px; line-height: 29.75px; margin: auto; padding: 0px; text-align: center;"><tbody style="margin: 0px; padding: 0px;" valign="top">
<tr style="margin: 0px; padding: 0px;"><td style="border-bottom-style: solid; border-bottom-width: 1px; border-left-width: 0px !important; border-right-style: solid; border-right-width: 1px; border-top-width: 0px !important; line-height: 1em; margin: auto; padding: 0.5em 1em; text-align: left !important; vertical-align: top;">Interval</td><td style="border-bottom-style: solid; border-bottom-width: 1px; border-left-width: 0px !important; border-right-style: solid; border-right-width: 1px; border-top-width: 0px !important; line-height: 1em; margin: auto; padding: 0.5em 1em; text-align: left !important; vertical-align: top;">Frequency</td><td style="border-bottom-style: solid; border-bottom-width: 1px; border-left-width: 0px !important; border-right-width: 0px !important; border-top-width: 0px !important; line-height: 1em; margin: auto; padding: 0.5em 1em; text-align: left !important; vertical-align: top;">Cumulative frequency</td></tr>
<tr style="margin: 0px; padding: 0px;"><td style="border-bottom-style: solid; border-bottom-width: 1px; border-left-width: 0px !important; border-right-style: solid; border-right-width: 1px; border-top-width: 0px !important; line-height: 1em; margin: auto; padding: 0.5em 1em; text-align: left !important; vertical-align: top;"><img src="http://m.everythingmaths.co.za/@@mobile_image?key=98effd93cb9c3f8a36df0c283e959bcd.png" style="border: 0px; margin: 0px; padding: 0px; vertical-align: middle;" /></td><td style="border-bottom-style: solid; border-bottom-width: 1px; border-left-width: 0px !important; border-right-style: solid; border-right-width: 1px; border-top-width: 0px !important; line-height: 1em; margin: auto; padding: 0.5em 1em; text-align: left !important; vertical-align: top;">5</td><td style="border-bottom-style: solid; border-bottom-width: 1px; border-left-width: 0px !important; border-right-width: 0px !important; border-top-width: 0px !important; line-height: 1em; margin: auto; padding: 0.5em 1em; text-align: left !important; vertical-align: top;"> </td></tr>
<tr style="margin: 0px; padding: 0px;"><td style="border-bottom-style: solid; border-bottom-width: 1px; border-left-width: 0px !important; border-right-style: solid; border-right-width: 1px; border-top-width: 0px !important; line-height: 1em; margin: auto; padding: 0.5em 1em; text-align: left !important; vertical-align: top;"><img src="http://m.everythingmaths.co.za/@@mobile_image?key=30470d5645c7bd471c308ae769599d9f.png" style="border: 0px; margin: 0px; padding: 0px; vertical-align: middle;" /></td><td style="border-bottom-style: solid; border-bottom-width: 1px; border-left-width: 0px !important; border-right-style: solid; border-right-width: 1px; border-top-width: 0px !important; line-height: 1em; margin: auto; padding: 0.5em 1em; text-align: left !important; vertical-align: top;">7</td><td style="border-bottom-style: solid; border-bottom-width: 1px; border-left-width: 0px !important; border-right-width: 0px !important; border-top-width: 0px !important; line-height: 1em; margin: auto; padding: 0.5em 1em; text-align: left !important; vertical-align: top;"> </td></tr>
<tr style="margin: 0px; padding: 0px;"><td style="border-bottom-style: solid; border-bottom-width: 1px; border-left-width: 0px !important; border-right-style: solid; border-right-width: 1px; border-top-width: 0px !important; line-height: 1em; margin: auto; padding: 0.5em 1em; text-align: left !important; vertical-align: top;"><img src="http://m.everythingmaths.co.za/@@mobile_image?key=569267538e96a336f499a3471c7e38f0.png" style="border: 0px; margin: 0px; padding: 0px; vertical-align: middle;" /></td><td style="border-bottom-style: solid; border-bottom-width: 1px; border-left-width: 0px !important; border-right-style: solid; border-right-width: 1px; border-top-width: 0px !important; line-height: 1em; margin: auto; padding: 0.5em 1em; text-align: left !important; vertical-align: top;">12</td><td style="border-bottom-style: solid; border-bottom-width: 1px; border-left-width: 0px !important; border-right-width: 0px !important; border-top-width: 0px !important; line-height: 1em; margin: auto; padding: 0.5em 1em; text-align: left !important; vertical-align: top;"> </td></tr>
<tr style="margin: 0px; padding: 0px;"><td style="border-bottom-style: solid; border-bottom-width: 1px; border-left-width: 0px !important; border-right-style: solid; border-right-width: 1px; border-top-width: 0px !important; line-height: 1em; margin: auto; padding: 0.5em 1em; text-align: left !important; vertical-align: top;"><img src="http://m.everythingmaths.co.za/@@mobile_image?key=e0ba894b9ac8456b4b4991ef2c992a6c.png" style="border: 0px; margin: 0px; padding: 0px; vertical-align: middle;" /></td><td style="border-bottom-style: solid; border-bottom-width: 1px; border-left-width: 0px !important; border-right-style: solid; border-right-width: 1px; border-top-width: 0px !important; line-height: 1em; margin: auto; padding: 0.5em 1em; text-align: left !important; vertical-align: top;">10</td><td style="border-bottom-style: solid; border-bottom-width: 1px; border-left-width: 0px !important; border-right-width: 0px !important; border-top-width: 0px !important; line-height: 1em; margin: auto; padding: 0.5em 1em; text-align: left !important; vertical-align: top;"> </td></tr>
<tr style="margin: 0px; padding: 0px;"><td style="border-bottom-width: 0px !important; border-left-width: 0px !important; border-right-style: solid; border-right-width: 1px; border-top-width: 0px !important; line-height: 1em; margin: auto; padding: 0.5em 1em; text-align: left !important; vertical-align: top;"><img src="http://m.everythingmaths.co.za/@@mobile_image?key=1420d7ac83c6f3aa0527edb826e18f3a.png" style="border: 0px; margin: 0px; padding: 0px; vertical-align: middle;" /></td><td style="border-bottom-width: 0px !important; border-left-width: 0px !important; border-right-style: solid; border-right-width: 1px; border-top-width: 0px !important; line-height: 1em; margin: auto; padding: 0.5em 1em; text-align: left !important; vertical-align: top;">6</td><td style="border: 0px !important; line-height: 1em; margin: auto; padding: 0.5em 1em; text-align: left !important; vertical-align: top;"> </td></tr>
</tbody></table>
</div>
<div class="problem" style="line-height: 29.75px; margin: 1.2em 0px; padding: 0px;">
<div class="example" style="border: 5px solid rgb(204, 204, 204); margin: 1.2em 0px; padding: 0px 0.5em;">
<div class="solution" style="line-height: 1.55em; margin: 0.5em 0px 0.5em 3em; padding: 0px;">
<h3 class="solution-header" style="color: #111111; font-size: 1.1em; margin: 0px; padding: 0px;">
<span class="cnx_label" style="margin: 0px; padding: 0px;">Solution</span></h3>
<step style="margin: 0px; padding: 0px;" xmlns="http://cnx.rice.edu/cnxml"><strong class="title" style="margin: 0px; padding: 0px;" xmlns="http://www.w3.org/1999/xhtml">Compute cumulative frequencies</strong><div class="para" style="line-height: 1.55em; margin-bottom: 1.2em; margin-top: 1.2em; padding: 0px;" xmlns="http://www.w3.org/1999/xhtml">
To determine the cumulative frequency, we add up the frequencies going down the table. The first cumulative frequency is just the same as the frequency, because we are adding it to zero. The final cumulative frequency is always equal to the sum of all the frequencies. This gives the following table:</div>
<div class="table" style="line-height: 1.55em; margin: 1.5em 0px; padding: 0px; text-align: center; width: 694.03125px;" xmlns="http://www.w3.org/1999/xhtml">
<table align="center" cellpadding="0" cellspacing="0" style="border-collapse: collapse; border-spacing: 0px; border: 1px solid; margin: auto; padding: 0px; text-align: center;"><caption align="bottom" class="table-text" style="margin: 0px; padding: 0.3em 0px 0px; text-align: left;"><span class="cnx_label" style="margin: 0px; padding: 0px;">Table 2</span></caption><tbody style="margin: 0px; padding: 0px;" valign="top">
<tr style="margin: 0px; padding: 0px;"><td style="border-bottom-style: solid; border-bottom-width: 1px; border-left-width: 0px !important; border-right-style: solid; border-right-width: 1px; border-top-width: 0px !important; line-height: 1em; margin: auto; padding: 0.5em 1em; text-align: left !important; vertical-align: top;">Interval</td><td style="border-bottom-style: solid; border-bottom-width: 1px; border-left-width: 0px !important; border-right-style: solid; border-right-width: 1px; border-top-width: 0px !important; line-height: 1em; margin: auto; padding: 0.5em 1em; text-align: left !important; vertical-align: top;">Frequency</td><td style="border-bottom-style: solid; border-bottom-width: 1px; border-left-width: 0px !important; border-right-width: 0px !important; border-top-width: 0px !important; line-height: 1em; margin: auto; padding: 0.5em 1em; text-align: left !important; vertical-align: top;">Cumulative frequency</td></tr>
<tr style="margin: 0px; padding: 0px;"><td style="border-bottom-style: solid; border-bottom-width: 1px; border-left-width: 0px !important; border-right-style: solid; border-right-width: 1px; border-top-width: 0px !important; line-height: 1em; margin: auto; padding: 0.5em 1em; text-align: left !important; vertical-align: top;"><img src="http://m.everythingmaths.co.za/@@mobile_image?key=98effd93cb9c3f8a36df0c283e959bcd.png" style="border: 0px; margin: 0px; padding: 0px; vertical-align: middle;" /></td><td style="border-bottom-style: solid; border-bottom-width: 1px; border-left-width: 0px !important; border-right-style: solid; border-right-width: 1px; border-top-width: 0px !important; line-height: 1em; margin: auto; padding: 0.5em 1em; text-align: left !important; vertical-align: top;">5</td><td style="border-bottom-style: solid; border-bottom-width: 1px; border-left-width: 0px !important; border-right-width: 0px !important; border-top-width: 0px !important; line-height: 1em; margin: auto; padding: 0.5em 1em; text-align: left !important; vertical-align: top;">5</td></tr>
<tr style="margin: 0px; padding: 0px;"><td style="border-bottom-style: solid; border-bottom-width: 1px; border-left-width: 0px !important; border-right-style: solid; border-right-width: 1px; border-top-width: 0px !important; line-height: 1em; margin: auto; padding: 0.5em 1em; text-align: left !important; vertical-align: top;"><img src="http://m.everythingmaths.co.za/@@mobile_image?key=30470d5645c7bd471c308ae769599d9f.png" style="border: 0px; margin: 0px; padding: 0px; vertical-align: middle;" /></td><td style="border-bottom-style: solid; border-bottom-width: 1px; border-left-width: 0px !important; border-right-style: solid; border-right-width: 1px; border-top-width: 0px !important; line-height: 1em; margin: auto; padding: 0.5em 1em; text-align: left !important; vertical-align: top;">7</td><td style="border-bottom-style: solid; border-bottom-width: 1px; border-left-width: 0px !important; border-right-width: 0px !important; border-top-width: 0px !important; line-height: 1em; margin: auto; padding: 0.5em 1em; text-align: left !important; vertical-align: top;">12</td></tr>
<tr style="margin: 0px; padding: 0px;"><td style="border-bottom-style: solid; border-bottom-width: 1px; border-left-width: 0px !important; border-right-style: solid; border-right-width: 1px; border-top-width: 0px !important; line-height: 1em; margin: auto; padding: 0.5em 1em; text-align: left !important; vertical-align: top;"><img src="http://m.everythingmaths.co.za/@@mobile_image?key=569267538e96a336f499a3471c7e38f0.png" style="border: 0px; margin: 0px; padding: 0px; vertical-align: middle;" /></td><td style="border-bottom-style: solid; border-bottom-width: 1px; border-left-width: 0px !important; border-right-style: solid; border-right-width: 1px; border-top-width: 0px !important; line-height: 1em; margin: auto; padding: 0.5em 1em; text-align: left !important; vertical-align: top;">12</td><td style="border-bottom-style: solid; border-bottom-width: 1px; border-left-width: 0px !important; border-right-width: 0px !important; border-top-width: 0px !important; line-height: 1em; margin: auto; padding: 0.5em 1em; text-align: left !important; vertical-align: top;">24</td></tr>
<tr style="margin: 0px; padding: 0px;"><td style="border-bottom-style: solid; border-bottom-width: 1px; border-left-width: 0px !important; border-right-style: solid; border-right-width: 1px; border-top-width: 0px !important; line-height: 1em; margin: auto; padding: 0.5em 1em; text-align: left !important; vertical-align: top;"><img src="http://m.everythingmaths.co.za/@@mobile_image?key=e0ba894b9ac8456b4b4991ef2c992a6c.png" style="border: 0px; margin: 0px; padding: 0px; vertical-align: middle;" /></td><td style="border-bottom-style: solid; border-bottom-width: 1px; border-left-width: 0px !important; border-right-style: solid; border-right-width: 1px; border-top-width: 0px !important; line-height: 1em; margin: auto; padding: 0.5em 1em; text-align: left !important; vertical-align: top;">10</td><td style="border-bottom-style: solid; border-bottom-width: 1px; border-left-width: 0px !important; border-right-width: 0px !important; border-top-width: 0px !important; line-height: 1em; margin: auto; padding: 0.5em 1em; text-align: left !important; vertical-align: top;">34</td></tr>
<tr style="margin: 0px; padding: 0px;"><td style="border-bottom-width: 0px !important; border-left-width: 0px !important; border-right-style: solid; border-right-width: 1px; border-top-width: 0px !important; line-height: 1em; margin: auto; padding: 0.5em 1em; text-align: left !important; vertical-align: top;"><img src="http://m.everythingmaths.co.za/@@mobile_image?key=1420d7ac83c6f3aa0527edb826e18f3a.png" style="border: 0px; margin: 0px; padding: 0px; vertical-align: middle;" /></td><td style="border-bottom-width: 0px !important; border-left-width: 0px !important; border-right-style: solid; border-right-width: 1px; border-top-width: 0px !important; line-height: 1em; margin: auto; padding: 0.5em 1em; text-align: left !important; vertical-align: top;">6</td><td style="border: 0px !important; line-height: 1em; margin: auto; padding: 0.5em 1em; text-align: left !important; vertical-align: top;">40</td></tr>
</tbody></table>
</div>
</step><step style="margin: 0px; padding: 0px;" xmlns="http://cnx.rice.edu/cnxml"><strong class="title" style="margin: 0px; padding: 0px;" xmlns="http://www.w3.org/1999/xhtml">Plot the ogive</strong><div class="para" style="line-height: 1.55em; margin-bottom: 1.2em; margin-top: 1.2em; padding: 0px;" xmlns="http://www.w3.org/1999/xhtml">
The first coordinate in the plot always starts at a <img src="http://m.everythingmaths.co.za/@@mobile_image?key=85067ce783e2f89ced535d722b824af5.png" style="border: 0px; margin: 0px; padding: 0px; vertical-align: middle;" />-value of 0 because we always start from a count of zero. So, the first coordinate is at <img src="http://m.everythingmaths.co.za/@@mobile_image?key=fac9193a2d9ea3bc802b53742dfdbace.png" style="border: 0px; margin: 0px; padding: 0px; vertical-align: middle;" /> — at the beginning of the first interval. The second coordinate is at the end of the first interval (which is also the beginning of the second interval) and at the first cumulative count, so <img src="http://m.everythingmaths.co.za/@@mobile_image?key=1fe0b90a67d48c8f4fc8a2ed1a175b99.png" style="border: 0px; margin: 0px; padding: 0px; vertical-align: middle;" />. The third coordinate is at the end of the second interval and at the second cumulative count, namely <img src="http://m.everythingmaths.co.za/@@mobile_image?key=aff31c49caa58b725e98116d7b0c03de.png" style="border: 0px; margin: 0px; padding: 0px; vertical-align: middle;" />, and so on.</div>
<div class="para" style="line-height: 1.55em; margin-bottom: 1.2em; margin-top: 1.2em; padding: 0px;" xmlns="http://www.w3.org/1999/xhtml">
Computing all the coordinates and connecting them with straight lines gives the following ogive.</div>
<div class="media" style="line-height: 1.55em; margin: 1.2em 0px; padding: 0px;" xmlns="http://www.w3.org/1999/xhtml">
<span class="media" id="" style="margin: 0px; padding: 0px;"><a href="http://m.everythingmaths.co.za/grade-11/11-statistics/tikzpictures/f80459166e4f49299fd057b182f2e305.png" style="border-bottom-color: rgb(204, 204, 204); border-bottom-style: solid; border-width: 0px 0px 0.1em; color: #205c90; margin: 0px; outline: none; padding: 0px; text-decoration: none;"><img alt="Image" class="mobile-resized" src="http://m.everythingmaths.co.za/@@shortimageurl/B94MD" style="border: none; float: none; margin: 0px; padding: 0px; vertical-align: middle;" /></a></span></div>
</step></div>
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<div class="para" style="margin-bottom: 1.2em; margin-top: 1.2em; padding: 0px;">
Ogives do look similar to frequency polygons, which we saw earlier. The most important difference between them is that an ogive is a plot of <em class="emphasis" style="margin: 0px; padding: 0px;">cumulative</em> values, whereas a frequency polygon is a plot of the values themselves. So, to get from a frequency polygon to an ogive, we would add up the counts as we move from left to right in the graph.</div>
<div class="para" style="margin-bottom: 1.2em; margin-top: 1.2em; padding: 0px;">
Ogives are useful for determining the median, percentiles and five number summary of data. Remember that the median is simply the value in the middle when we order the data. A quartile is simply a quarter of the way from the beginning or the end of an ordered data set. With an ogive we already know how many data values are above or below a certain point, so it is easy to find the middle or a quarter of the data set.</div>
<div class="para" style="margin-bottom: 1.2em; margin-top: 1.2em; padding: 0px;">
source- m.everythingmaths.co.za</div>
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Unknownnoreply@blogger.com0tag:blogger.com,1999:blog-3230242783536988080.post-55672137931694126622013-07-07T00:41:00.002-07:002013-07-07T00:43:00.461-07:00STATISTICS (MODE AND MEAN)<div dir="ltr" style="text-align: left;" trbidi="on">
<span style="font-size: x-large;">MODE</span><br />
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<span style="font-size: x-large;">When individual series or discrete series are given then mode is the most frequently occuring observation.But when continuous series are given then we have to take out the average mode that is</span></div>
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<span style="font-size: x-large;"><span style="color: #e06666;">mode</span> =</span><span style="background-color: white; color: #333333; font-family: arial, helvetica, clean, sans-serif; line-height: 16px;"><span style="font-size: large;"> L + [(f1-f0) / (2f1-f0-f2)] x h</span></span></div>
<span style="background-color: white; color: #333333; font-family: arial, helvetica, clean, sans-serif; font-size: large; line-height: 16px;"></span><br />
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<span style="background-color: white; color: #333333; font-family: arial, helvetica, clean, sans-serif; font-size: large; line-height: 16px;"><br /></span></div>
<span style="background-color: white; color: #333333; font-family: arial, helvetica, clean, sans-serif; font-size: large; line-height: 16px;">
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where:</div>
</span><span style="font-size: large;"><span style="background-color: white; color: #333333; font-family: arial, helvetica, clean, sans-serif; line-height: 16px;"><div>
<span style="font-size: large;"><span style="background-color: white; color: #333333; font-family: arial, helvetica, clean, sans-serif; line-height: 16px;"><br /></span></span></div>
L is the lower class limit of modal class</span><br style="background-color: white; color: #333333; font-family: arial, helvetica, clean, sans-serif; line-height: 16px;" /><span style="background-color: white; color: #333333; font-family: arial, helvetica, clean, sans-serif; line-height: 16px;"><br /></span></span><br />
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<span style="font-size: large;"><span style="background-color: white; color: #333333; font-family: arial, helvetica, clean, sans-serif; line-height: 16px;">f0 is the frequency previous to the model class</span></span><br />
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<span style="font-size: large;"><span style="background-color: white; color: #333333; font-family: arial, helvetica, clean, sans-serif; line-height: 16px;"><br /></span></span></div>
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<span style="font-size: large;"><span style="background-color: white; color: #333333; font-family: arial, helvetica, clean, sans-serif; line-height: 16px;">f1</span><span style="background-color: white; color: #333333; font-family: arial, helvetica, clean, sans-serif; line-height: 16px;"> is the Frequency of the model class</span><br style="background-color: white; color: #333333; font-family: arial, helvetica, clean, sans-serif; line-height: 16px;" /><span style="background-color: white; color: #333333; font-family: arial, helvetica, clean, sans-serif; line-height: 16px;"><br /></span></span></div>
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<span style="background-color: white; color: #333333; font-family: arial, helvetica, clean, sans-serif; font-size: large; line-height: 16px;">f2 is the frequency next to the model class</span></div>
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<span style="font-size: large;"><span style="background-color: white; color: #333333; font-family: arial, helvetica, clean, sans-serif; line-height: 16px;"><br /></span></span></div>
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<span style="background-color: white; color: #333333; font-family: arial, helvetica, clean, sans-serif; font-size: large; line-height: 16px;">h is the size of model class i.e. difference between upper </span></div>
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<span style="background-color: white; color: #333333; font-family: arial, helvetica, clean, sans-serif; font-size: large; line-height: 16px;"><br /></span></div>
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<span style="background-color: white; color: #333333; font-family: arial, helvetica, clean, sans-serif; font-size: large; line-height: 16px;">and lower class boundaries of model class.</span></div>
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<span style="font-size: large;"><span style="background-color: white; color: #333333; font-family: arial, helvetica, clean, sans-serif; line-height: 16px;"><br /></span></span></div>
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<span style="font-size: large;"><span style="background-color: white; color: #333333; font-family: arial, helvetica, clean, sans-serif; line-height: 16px;">Model class is a class with the maximum frequency.</span></span><br />
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<span style="font-size: x-large;"><br /></span></div>
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<span style="font-size: x-large;">MEDIAN</span><span style="font-size: x-large;">Same is the case when we have to find median.There are different methods to find median in different series.When individual or discrete series are given then </span></div>
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<span style="font-size: x-large;">if no. of observations are odd then</span></div>
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<span style="font-size: x-large;"> <span style="color: #cc0000;">median</span> = (n+1/2)th observations.</span></div>
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<span style="font-size: x-large;">if no. of observations are even then</span></div>
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<span style="font-size: x-large;"><span style="color: #990000;">median</span>=mean of (n/2)th and (n/2</span><span style="background-color: white; font-family: arial, helvetica, 'bitstream vera sans', clean, sans-serif; font-size: large; line-height: 16px;">÷</span><span style="font-size: x-large;">1)th observations.</span></div>
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<span style="font-size: x-large;"><br /></span></div>
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<span style="font-size: x-large;">When continuous series are given then </span></div>
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<span style="font-size: x-large;"><span style="color: #660000;">median</span>= L+(n/2-c.f.</span><span style="font-size: large;"><span style="background-color: white; font-family: arial, helvetica, 'bitstream vera sans', clean, sans-serif; line-height: 16px;">÷f)</span><span style="background-color: white; font-family: arial, helvetica, 'bitstream vera sans', clean, sans-serif; line-height: 16px;"> </span><span style="background-color: white; font-family: arial, helvetica, 'bitstream vera sans', clean, sans-serif; line-height: 16px;">×h</span></span></div>
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<span style="font-size: x-large;"> </span></div>
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Unknownnoreply@blogger.com0tag:blogger.com,1999:blog-3230242783536988080.post-49405429238753634022013-07-04T21:11:00.004-07:002013-07-04T21:12:46.750-07:00STATISTICS OVERVIEW (MEAN)<div dir="ltr" style="text-align: left;" trbidi="on">
<span style="background-color: white;">THERE ARE THREE TYPES OF SERIES-</span><br />
<span style="background-color: white;">1.INDIVIDUAL SERIES</span><br />
<span style="background-color: white;">- Let x1, x2.....................xn be given obs. and 'n' be the total no. of obs.,then <span style="font-family: Tahoma, Arial; font-size: 11px;"> </span></span><br />
<span style="background-color: white; font-family: Tahoma, Arial;">mean = Σxi/n</span><br />
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<span style="background-color: white;"> where xi is sum of all obs.</span></div>
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<span style="background-color: white;"> </span></div>
<div align="justify" style="font-family: Tahoma, Arial; margin: 0px; padding: 0px;">
<span style="background-color: white;">2.DISCRETE SERIES</span></div>
<div align="justify" style="font-family: Tahoma, Arial; margin: 0px; padding: 0px;">
<span style="background-color: white;">- Let x1,x2..........xn be given obs. and f1,f2.....fn be their frequencies then </span></div>
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<span style="background-color: white;">mean=<span style="font-size: 11px;"> </span>Σfixi/Σfi</span></div>
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<span style="background-color: white; font-size: 11px;"><br /></span></div>
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<span style="background-color: white; font-size: 11px;"><br /></span></div>
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<span style="background-color: white;">3.CONTINUOUS SERIES</span></div>
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<span style="background-color: white;">- Continuous series means when class intervals are given.</span></div>
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<span style="background-color: white;"><br /></span></div>
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<span style="background-color: white;">Mean can be found by three methods here-</span></div>
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<span style="background-color: white;"><br /></span></div>
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<span style="background-color: white;">(A)DIRECT METHOD</span></div>
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<span style="background-color: white;">When upper and lower limits of the class intervals are small.</span></div>
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<span style="background-color: white;"><br /></span></div>
<div align="justify" style="font-family: Tahoma, Arial; margin: 0px; padding: 0px;">
<span style="background-color: white;">mean=Σfixi/Σfi </span></div>
<div align="justify" style="font-family: Tahoma, Arial; margin: 0px; padding: 0px;">
<span style="background-color: white;"><br /></span></div>
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<span style="background-color: white;"><br /></span></div>
<div align="justify" style="font-family: Tahoma, Arial; margin: 0px; padding: 0px;">
<span style="background-color: white;">(B)DEVIATION OR ASSUMED MEAN METHOD</span></div>
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<span style="background-color: white;">When upper and lower limits of the class intervals are large,then this method is used.</span></div>
<div align="justify" style="font-family: Tahoma, Arial; margin: 0px; padding: 0px;">
<span style="background-color: white;">the assumed mean is taken from class mark(xi)</span></div>
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<span style="background-color: white;"><br /></span></div>
<div align="justify" style="font-family: Tahoma, Arial; margin: 0px; padding: 0px;">
<span style="background-color: white;">mean=<span style="text-align: left;"> </span><span style="text-align: left;">A + Σfidi/Σfi, where d = mean – A, where A is the assumed mean.</span></span></div>
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<span style="background-color: white; text-align: left;"><br /></span></div>
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<span style="background-color: white; text-align: left;"><br /></span></div>
<div align="justify" style="font-family: Tahoma, Arial; margin: 0px; padding: 0px;">
<span style="background-color: white; text-align: left;">(C)STEP DEVIATION METHOD</span></div>
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<span style="background-color: white; text-align: left;">This method is used to make the calculations for finding the mean easy.</span></div>
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<span style="background-color: white; text-align: left;"><br /></span></div>
<div align="justify" style="font-family: Tahoma, Arial; margin: 0px; padding: 0px;">
<span style="background-color: white;"><span style="text-align: left;">Mean=</span><span style="text-align: left;"> </span><span style="text-align: left;">A + (Σfiui/Σfi)h, where A is the assumed mean and u = di/h.</span></span></div>
<div align="justify" style="font-family: Tahoma, Arial; margin: 0px; padding: 0px;">
<span style="background-color: white; text-align: left;"><br /></span></div>
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<span style="background-color: white; text-align: left;"><br /></span></div>
<div align="justify" style="font-family: Tahoma, Arial; margin: 0px; padding: 0px;">
<span style="background-color: white;"><span style="text-align: left;">NOTE- Step Deviation method is used mainly for finding the mean when data is given in the continuous series with large value of upper and lower limits.We can use the other two methods if they are specified in the question or according to the conditions of the questions.</span><span style="text-align: left;"> </span></span></div>
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<span style="font-size: 11px;"><br /></span></div>
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Unknownnoreply@blogger.com0tag:blogger.com,1999:blog-3230242783536988080.post-45875854504335326142013-05-27T06:19:00.006-07:002013-05-27T06:19:59.323-07:00INTRODUCTION TO TRIGNOMETRY<div dir="ltr" style="text-align: left;" trbidi="on">
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<iframe allowfullscreen="" frameborder="0" height="315" src="http://www.youtube.com/embed/nf69QEuQ1IM" width="560"></iframe><br />
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Source : exam fear.com<br />
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Unknownnoreply@blogger.com0tag:blogger.com,1999:blog-3230242783536988080.post-82353824113095457142013-05-12T03:54:00.002-07:002013-05-12T03:54:32.863-07:00MATHEMAGIC 2!!<div dir="ltr" style="text-align: left;" trbidi="on">
<iframe allowfullscreen="" frameborder="0" height="315" src="http://www.youtube.com/embed/_dpS-FYin1A" width="420"></iframe><br />
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Unknownnoreply@blogger.com0tag:blogger.com,1999:blog-3230242783536988080.post-37410143483688362082013-05-12T03:51:00.002-07:002013-05-12T03:51:59.007-07:00MATHEMAGIC!!<div dir="ltr" style="text-align: left;" trbidi="on">
<iframe allowfullscreen="" frameborder="0" height="315" src="http://www.youtube.com/embed/BcyU4kS-SDA" width="420"></iframe><br />
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Unknownnoreply@blogger.com0tag:blogger.com,1999:blog-3230242783536988080.post-19626191349279617112013-05-12T03:46:00.001-07:002013-05-12T03:46:40.803-07:00MATHS TRICKS FOR DIVISION<div dir="ltr" style="text-align: left;" trbidi="on">
<iframe allowfullscreen="" frameborder="0" height="315" src="http://www.youtube.com/embed/9rzhGphfjWA" width="420"></iframe><br />
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